1. Lecture 2: Ultracold fermions Fermions in optical lattices. Fermi Hubbard model. Current state of experiments Lattice modulation experiments Doublon lifetimes Stoner instability

2. Ultracold fermions in optical lattices

3. t U t Fermionic atoms in optical lattices Experiments with fermions in optical lattice, Kohl et al., PRL 2005

4. Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model Quantum simulations with ultracold atoms

5. Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave superconductor - ?

6. Repulsive Hubbard model at half-filling Mott state without AF order current experiments TNTN U

7. Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies R. Joerdens et al., Nature (2008) Compressibility measurements U. Schneider et al., Science (2008)

8. Lattice modulation experiments with fermions in optical lattice. Related theory work: Kollath et al., PRL (2006) Huber, Ruegg, PRB (2009) Orso, Iucci, et al., PRA (2009) Probing the Mott state of fermions Sensarma, Pekker, Lukin, Demler, PRL (2009)

9. Lattice modulation experiments Probing dynamics of the Hubbard model Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Modulate lattice potential Doubly occupied sites created when frequency w matches Hubbard U

10. Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., Nature 455:204 (2008)

11. Mott state Regime of strong interactions U>>t. Mott gap for the charge forms at Antiferromagnetic ordering at “High” temperature regime “Low” temperature regime All spin configurations are equally likely. Can neglect spin dynamics. Spins are antiferromagnetically ordered or have strong correlations

12. Schwinger bosons and Slave Fermions BosonsFermions Constraint : Singlet Creation Boson Hopping

13. Schwinger bosons and slave fermions Fermion hopping Doublon production due to lattice modulation perturbation Second order perturbation theory. Number of doublons Propagation of holes and doublons is coupled to spin excitations. Neglect spontaneous doublon production and relaxation.

14. d h Assume independent propagation of hole and doublon (neglect vertex corrections) =+ Self-consistent Born approximation Schmitt-Rink et al (1988), Kane et al. (1989) Spectral function for hole or doublon Sharp coherent part: dispersion set by t 2 /U, weight by t/U Incoherent part: dispersion Propagation of holes and doublons strongly affected by interaction with spin waves Schwinger bosons Bose condensed “Low” Temperature Oscillations reflect shake-off processes of spin waves

15. “Low” Temperature Rate of doublon production Sharp absorption edge due to coherent quasiparticles Broad continuum due to incoherent part Spin wave shake-off peaks

16. “High” Temperature Calculate self-energy of doublons and holes interacting with incoherent spin excitations (Schwinger bosons) in the non-crossing approximation Equivalent to Retraceable Path Approximation Brinkmann & Rice, 1970 Sensarma et al., PRL (2009) Tokuno et al. arXIv:1106.1333 In calculating spectral function consider paths with no closed loops

17. Spectral Fn. of single hole Experiment: R. Joerdens et al., Nature 455:204 (2008) Theory: Sensarma et al., PRL (2009) Tokuno et al. arXIv:1106.1333 “High” Temperature d h Doublon production rate

18. Reduced probability to find a singlet on neighboring sites Radius Density P singlet D. Pekker et al., upublished Temperature dependence

19. Fermions in optical lattice. Decay of repulsively bound pairs Ref: N. Strohmaier et al., PRL 2010 Experiments: ETH group Theory: Sensarma, Pekker, et. al.

20. Fermions in optical lattice. Decay of repulsively bound pairs Doublons – repulsively bound pairs What is their lifetime? Excess energy U should be converted to kinetic energy of single atoms Decay of doublon into a pair of quasiparticles requires creation of many particle-hole pairs Direct decay is not allowed by energy conservation

21. Fermions in optical lattice. Decay of repulsively bound pairs Experiments: N. Strohmaier et. al.

22.  Energy carried by spin excitations ~ J =4t 2 /U  Relaxation requires creation of ~U 2 /t 2 spin excitations Relaxation of doublon- hole pairs in the Mott state Energy U needs to be absorbed by spin excitations Relaxation rate Very slow, not relevant for ETH experiments Sensarma et al., PRL 2011

23. Doublon decay in a compressible state Excess energy U is converted to kinetic energy of single atoms Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs U p-pp-p p-hp-h p-hp-h p-hp-h

24. Doublon decay in a compressible state Perturbation theory to order n=U/6t Decay probability Doublon PropagatorInteracting “Single” Particles

25. Doublon decay in a compressible state To calculate the rate: consider processes which maximize the number of particle-hole excitations Expt: ETHZ Theory: Harvard N. Strohmaier et al., PRL 2010

26. Why understanding doublon decay rate is interesting Important for adiabatic preparation of strongly correlated systems in optical lattices Response functions of strongly correlated systems at high frequencies. Important for numerical analysis. Prototype of decay processes with emission of many interacting particles. Example: resonance in nuclear physics: (i.e. delta-isobar) Analogy to pump and probe experiments in condensed matter systems

27. Doublon relaxation in organic Mott insulators ET-F 2 TCNQ

28. One dimensional Mott insulator ET-F 2 TCNQ t=0.1 eV U=0.7 eV

29. Surprisingly long relaxation time 840 fs h/t = 40 fs H. Okamoto et al., PRL 98:37401 (2007) S. Wall et al. Nature Physics 7:114 (2011) Photoinduced metallic state

30. H. Okamoto et al., PRL 98:37401 (2007) S. Wall et al. Nature Physics 7:114 (2011) Photoinduced metallic state t=0.1 eV w=4t=0.4eV U=0.7 eV 1400 fs comparable to experimentally measured 840 ms

31. Exploring beyond simple Hubbard model with ultracold fermions

32. SU(N) Hubbard model with Ultracold Alkaline-Earth Atoms Ex: 87 Sr (I = 9/2) |g  = 1 S 0 |e  = 3 P 0 698 nm 150 s ~ 1 mHz orbital degree of freedom ⇒ spin-orbital physics → Kugel-Khomskii model [transition metal oxides with perovskite structure] → SU(N) Kondo lattice model [for N=2, colossal magnetoresistance in manganese oxides and heavy fermion materials] Nuclear spin decoupled from electrons SU(N=2I+1) symmetry → SU(N) Hubbard models ⇒ valence-bond-solid & spin-liquid phases Theory: A. Gorshkov, et al., Nature Physics 2010 Experiments: realization of SU(6) fermions Takahashi et al. PRL (2010) Also J. Ye et al., Science (2011)

33. New dynamical symmetry: identical slowdown of expansion for attractive and repulsive interactions Nonequilibrium dynamics: expansion of interacting fermions in optical lattice U. Schneider et al., Nature Physics 2012

34. Competition between pairing and ferromagnetic instabilities in ultracold Fermi gases near Feshbach resonances Phys. Rev. Lett. 2010 D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L. Pollet, M. Zwierlein, E. Demler Motivated by experiments of G.-B. Jo et al., Science (2009)

35. then Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U N(0) = 1 U – interaction strength N(0) – density of states at Fermi level Theoretical proposals for observing Stoner instability with cold gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); … Kanamori’s counter-argument: renormalization of U. Recent work on hard sphere potentials: Pilati et al. (2010); Chang et al. (2010)

36. Experiments were done dynamically. What are implications of dynamics? Why spin domains could not be observed? Earlier work by C. Salomon et al., 2003

37. Is it sufficient to consider effective model with repulsive interactions when analyzing experiments? Feshbach physics beyond effective repulsive interaction

38. Feshbach resonance Interactions between atoms are intrinsically attractive Effective repulsion appears due to low energy bound states Example: scattering length V(x) V 0 tunable by the magnetic field Can tune through bound state

39. Feshbach resonance Two particle bound state formed in vacuum BCS instability Stoner instability Molecule formation and condensation This talk: Prepare Fermi state of weakly interacting atoms. Quench to the BEC side of Feshbach resonance. System unstable to both molecule formation and Stoner ferromagnetism. Which instability dominates ?

40. Pair formation

41. Two-particle scattering in vacuum k-k p -p Microscopic Hamiltonian Schrödinger equation

42. Lippman-Schwinger equation For positive scattering length bound state at appears as a pole in the T-matrix k k -k k -p’-p p pk p p’ -p T-matrix On-shell T-matrix. Universal low energy expression

43. Cooperon Two particle scattering in the presence of a Fermi sea k p -k -p Need to make sure that we do not include interaction effects on the Fermi liquid state in scattered state energy

44. Cooperon vs T-matrix k k -k k -p’-p p pk p p’ -p

45. Cooper channel response function Linear response theory Induced pairing field Response function Poles of the Cooper channel response function are given by

46. Cooper channel response function Poles of the response function,, describe collective modes Linear response theory Time dependent dynamics When the mode frequency has negative imaginary part, the system is unstable

47. Pairing instability regularized BCS side Instability rate coincides with the equilibrium gap (Abrikosov, Gorkov, Dzyaloshinski) Instability to pairing even on the BEC side Related work: Lamacraft, Marchetti, 2008

48. Pairing instability Intuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance. Energy and momentum conservation laws can not be satisfied. This argument applies in vacuum. Fermi sea prevents formation of real Feshbach molecules by Pauli blocking. Molecule Fermi sea

49. Pairing instability Time dependent variational wavefunction Time dependence of u k (t) and v k (t) due to D BCS (t) For small D BCS (t):

50. Pairing instability From wide to narrow resonances Effects of finite temperature Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005 Observed in recent experiments by Grimm’s group, arXiv:1112.0020

51. Magnetic instability

52. Stoner instability. Naïve theory Linear response theory Spin response function Spin collective modes are given by the poles of response function Negative imaginary frequencies correspond to magnetic instability

53. RPA analysis for Stoner instability Self-consistent equation on response function RPA expression for the spin response function Spin susceptibility for non-interacting gas

54. Quench dynamics across Stoner instability Unstable modes determine characteristic lengthscale of magnetic domains For U>U c unstable collective modes Stoner criterion

55. Stoner quench dynamics in D=3 Growth rate of magnetic domains Domain size Unphysical divergence of the instability rate at unitarity Scaling near transition

56. Stoner instability Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea. Stoner instability is determined by two particle scattering amplitude =+++ … =++

57. Stoner instability RPA spin susceptibility Interaction = Cooperon

58. Stoner instability Pairing instability always dominates over pairing If ferromagnetic domains form, they form at large q

59. Spin fluctuations relative to noninteracting fermions Only short range correlations and no domain formation Extremely fast molecule formation rate at short times Atoms loss rate is 13% of E F on resonance (averaged over trap) Tests of the Stoner magnetism C. Sanner et al., arXiv:1108.2017

60. Lecture 2: Ultracold fermions Fermions in optical lattices. Fermi Hubbard model. Current state of experiments Lattice modulation experiments Doublon lifetimes Stoner instability

61. Future directions in ultracold atoms Long intrinsic time scales - Interaction energy and bandwidth ~ 1kHz - System parameters can be changed over this time scale Decoupling from external environment - Long coherence times Can achieve highly non equilibrium quantum many-body states Nonequilibrium quantum many-body dynamics

62. Do we have emergent universal phenomena in nonequilibrium dynamics of many-body quantum systems? Emergent phenomena in dynamics of classical systems Solitons in nonlinear wave propagation Bernard cells in the presence of T gradient Universality in quantum many- body systems in equilibrium Broken symmetries Fermi liquid state

64. Hubbard model at half filling U TNTN paramagnetic Mott phase Paramagnetic Mott phase: one fermion per site charge fluctuations suppressed no spin order BCS-type theory applies Heisenberg model applies

65. Hubbard model at half filling U TNTN BCS-type theory applies Heisenberg model applies current experiments